reserve x,y for Real,
  i, j for non zero Element of NAT,
  I, O for non empty set,
  s,s1,s2,s3 for Element of I,
  w, w1, w2 for FinSequence of I,
  t for Element of O,
  S for non empty FSM over I,
  q, q1 for State of S;

theorem Th5:
  (for w1, w2 st w1.1 = w2.1 holds
  GEN(w1, the InitS of S), GEN(w2, the InitS of S) are_c=-comparable)
  implies S is calculating_type
proof
  assume
A1: for w1, w2 st w1.1 = w2.1 holds
  GEN(w1, the InitS of S), GEN(w2, the InitS of S) are_c=-comparable;
  let j, w1, w2 such that
A2: w1.1 = w2.1 and
A3: j <= len(w1)+1 and
A4: j <= len(w2)+1;
A5: dom GEN(w1, the InitS of S) =
  Seg len(GEN(w1,the InitS of S)) by FINSEQ_1:def 3
    .= Seg(len w1 + 1) by FSM_1:def 2;
A6: dom GEN(w2, the InitS of S) =
  Seg len(GEN(w2,the InitS of S)) by FINSEQ_1:def 3
    .= Seg(len w2 + 1) by FSM_1:def 2;
A7: 1 <= j by NAT_1:14;
  then
A8: j in dom GEN(w1, the InitS of S) by A3,A5,FINSEQ_1:1;
  j in dom GEN(w2, the InitS of S) by A4,A6,A7,FINSEQ_1:1;
  then
A9: j in dom GEN(w1, the InitS of S) /\ dom GEN(w2, the InitS of S)
  by A8,XBOOLE_0:def 4;
  GEN(w1, the InitS of S), GEN(w2, the InitS of S) are_c=-comparable by A1,A2;
  then GEN(w1, the InitS of S) c= GEN(w2, the InitS of S) or
  GEN(w2, the InitS of S) c= GEN(w1, the InitS of S);
  then GEN(w1, the InitS of S) tolerates GEN(w2, the InitS of S)
  by PARTFUN1:54;
  hence thesis by A9,PARTFUN1:def 4;
end;
